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We introduce the tensor numerical method for solving optimal control problemsthat are constrained by fractional 2D and 3D elliptic operators with variablecoefficients. We solve the governing equation for the control function whichincludes a sum of the fractional operator and its inverse, both discretizedover large 3D $n\times n \times n$ spacial grids. Using the diagonalization ofthe arising matrix valued functions in the eigenbasis of the 1D Sturm-Liouvilleoperators, we construct the rank-structured tensor approximation withcontrollable precision for the discretized fractional elliptic operators andthe respective preconditioner. The right-hand side in the constraining equation(the optimal design function) is supposed to be represented in a form of alow-rank canonical tensor. Then the equation for the control function is solvedin a tensor structured format by using preconditioned CG iteration with theadaptive rank truncation procedure that also ensures the accuracy ofcalculations, given an $\varepsilon$-threshold. This method reduces thenumerical cost for solving the control problem to $O(n \log n)$ (plus thequadratic term $O(n^2)$ with a small weight), which is superior to theapproaches based on the traditional linear algebra tools that yield at least$O(n^3 \log n)$ complexity in the 3D case. The storage for the representationof all 3D nonlocal operators and functions involved is also estimated by $O(n\log n)$. This essentially outperforms the traditional methods operating withfully populated $n^3 \times n^3$ matrices and vectors in $\mathbb{R}^{n^3}$.Numerical tests for 2D/3D control problems indicate the almost linearcomplexity scaling of the rank truncated PCG iteration in the univariate gridsize $n$.

Tensor Method for Optimal Control Problems Constrained by Fractional 3D Elliptic Operator with Variable Coefficients